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MS#74. Endomorphisms and Visualization, 1992
- Subjects: Discrete dynamics
- Written: 21 December 1992
- Abstract. The emerging role of nonlinear dynamical
systems theory in the sciences, both in
model building and data analysis, is leading
to a uniform working strategy in all fields of
science. Thus, compatible models in these
fields may be combined into massively
complex models for whole systems, such
as Gaian physiology (land, ocean, and
atmosphere), human population growth
and demographics, the world economy,
or combinations of these. These massive
models, though simpler than nature, may be
too complex for our understanding. This
problem is the basis for the new emphasis
on scientific visualization
in general, and
dynamical visualization
in particular.
That is, given a continuous or discrete
dynamical system of very high dimension,
how can we visualize and understand its
behavior?
In this paper we will consider a special
case of this problem, in which the
massively complex dynamical system is
a semi-cascade, that is, the iteration
of non-invertible map, or
endomorphism.
The strategy of visualization of this
system consists of projection of the
trajectories onto a low-dimensional
(especially, two- or three- dimensional)
subspace. The new method of critical
curves, discovered by Christian Mira
in 1964 for the study of plane
endomorphisms, provides tools to infer
the behavior of the massive system
from the simple observation of its
projection onto a subspace.
- [PDF], 388 KB, 3 pages
Last revised
by Ralph Abraham
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